By now y’all know option traders have the ATM straddle approximation burned into their retina:
straddle ≈ .8 Sσ √T
A useful approximation I did not explain in the interview is the similar-looking ATM gamma formula for a Black-Scholes straddle:
Γ ≈ .8 / (Sσ√T)
The three things that shrink gamma are in the denominator:
Higher S (price): The same $1 move is a smaller percentage move on a more expensive underlying.
Higher σ (vol): The option is already “priced for action.” The curvature of the price function gets spread over a wider range of expected outcomes. More vol → flatter curvature near the money → less gamma.
Higher T (time): Same logic as vol. More time spreads the curvature out. The more time to expiry the less a given move influences the delta of the option. The delta of 10-year option is not going to change much based on how the underlying changes day-to-day.
A couple of educational points:
- Take note of the scaling. Double the vol, gamma roughly halves. You need to quadruple DTE to get the same effect.
- As always, a good habit when trying to understand greek levers, is to take examples to extremes. If you raise DTE or vol to infinity, all options go to their maximum value. For calls, that’s the spot price itself. For puts, it’s their strike price. That means calls go to 100% delta since they move dollar-for-dollar with the spot. Puts go to 0 delta. It doesn’t matter where the spot price goes, the option is already at its max value. It doesn’t change. If a call is 100% delta and a put is 0% delta, the option has no gamma. Its delta doesn’t change with respect to the spot.
Going back to those formulas for a moment:
straddle ≈ .8 Sσ √T
Γ ≈ .8 / (Sσ√T)
The denominator of gamma = straddle/.8
Substituting:
Γ ≈ .8 /(straddle/.8)
Γ ≈ .8 /(straddle/.8)
Γ ≈ .64 /straddle
So when you want to do mental math you take “2/3 of the inverse of the straddle.”
This might sound obtuse, but taking inverse or “1 over” some number should be one of the fastest mental math operations anyone dealing with investing does. After all, when you see any ratio like P/E or P/FCF you are immediately flipping that to a yield where it can be compared with things like interest rates or cap rates.
If a straddle is $5, the gamma is 2/3 of $.20 or ~.13
And we know that doubling the straddle halves the gamma so you can just memorize that a $10 straddle has ~6.6 cents of gamma and linearly estimate gamma for any straddle price relative to that (ie $20 straddle is about 3.3 cents of gamma and $15 straddle is in the middle of 3.3 and 6.6).
And of course there’s time scaling. To find an option that has double the gamma you need to cut the DTE by 1/4.
Keep flipping this stuff over in your head, it’s satisfying, and it thickens the myelin around whatever brain cells you sacrifice to options damage.